American Institute of Mathematical Sciences

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2015, 12(1): 83-97. doi: 10.3934/mbe.2015.12.83

Delayed population models with Allee effects and exploitation

Eduardo Liz 1, and  Alfonso Ruiz-Herrera 2,

1. 

Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36310 Vigo
2. 

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary

Received  April 2014 Revised  October 2014 Published  December 2014

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.
Keywords: bifurcation., bistability, difference equation, global stability, delay differential equation, Allee effect, Population model.
Mathematics Subject Classification: Primary: 34K18, 34K20, 92D25; Secondary: 37E0.
Citation: Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83
References:
[1]

D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084.  Google Scholar

[2]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87. doi: 10.1016/j.mbs.2013.12.003.  Google Scholar

[3]

C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990.  Google Scholar

[4]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.  Google Scholar

[5]

A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912. Google Scholar

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[7]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn., 4 (2010), 397-408. doi: 10.1080/17513750903377434.  Google Scholar

[8]

S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?, J. Math. Biol., 65 (2012), 1411-1415. doi: 10.1007/s00285-011-0496-3.  Google Scholar

[9]

K. P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18.  Google Scholar

[10]

C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114. doi: 10.1016/j.jde.2013.12.015.  Google Scholar

[11]

A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54.  Google Scholar

[12]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224.  Google Scholar

[13]

M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z.  Google Scholar

[14]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.  Google Scholar

[15]

T. Krisztin and E. Liz, Bubbles for a class of delay differential equations, Qual. Theory Dyn. Syst., 10 (2011), 169-196. doi: 10.1007/s12346-011-0055-8.  Google Scholar

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.  Google Scholar

[17]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221. doi: 10.1007/s12080-009-0064-2.  Google Scholar

[18]

E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.  Google Scholar

[19]

E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215.  Google Scholar

[20]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266. doi: 10.1016/j.jde.2013.08.007.  Google Scholar

[21]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222.  Google Scholar

[22]

J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128. doi: 10.1007/BF01790539.  Google Scholar

[23]

G. Röst, On the global attractivity controversy for a delay model of hematopoiesis, Appl. Math. Comput., 190 (2007), 846-850. doi: 10.1016/j.amc.2007.01.103.  Google Scholar

[24]

G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. doi: 10.1098/rspa.2007.1890.  Google Scholar

[25]

S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517. doi: 10.1007/1-4020-3647-7_11.  Google Scholar

[26]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260. doi: 10.1007/s002850000070.  Google Scholar

[27]

S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.  Google Scholar

[28]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-94-015-8897-3.  Google Scholar

[29]

A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993. doi: 10.1007/978-94-011-1763-0.  Google Scholar

[30]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.  Google Scholar

[31]

A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77. doi: 10.1016/j.mbs.2011.04.004.  Google Scholar

[32]

T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973. doi: 10.1098/rspa.2009.0650.  Google Scholar

show all references

References:
[1]

D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, J. Theoret. Biol., 218 (2002), 375-394. doi: 10.1006/jtbi.2002.3084.  Google Scholar

[2]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model, Math. Biosci., 248 (2014), 78-87. doi: 10.1016/j.mbs.2013.12.003.  Google Scholar

[3]

C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, Hoboken, New Jersey, 1990.  Google Scholar

[4]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.  Google Scholar

[5]

A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907-12912. Google Scholar

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[7]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, J. Biol. Dyn., 4 (2010), 397-408. doi: 10.1080/17513750903377434.  Google Scholar

[8]

S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?, J. Math. Biol., 65 (2012), 1411-1415. doi: 10.1007/s00285-011-0496-3.  Google Scholar

[9]

K. P. Hadeler, Neutral delay equations from and for population dynamics, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1-18.  Google Scholar

[10]

C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differential Equations, 256 (2014), 2101-2114. doi: 10.1016/j.jde.2013.12.015.  Google Scholar

[11]

A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54.  Google Scholar

[12]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164-224.  Google Scholar

[13]

M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect, Theor. Ecol., 7 (2014), 335-349. doi: 10.1007/s12080-014-0222-z.  Google Scholar

[14]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.  Google Scholar

[15]

T. Krisztin and E. Liz, Bubbles for a class of delay differential equations, Qual. Theory Dyn. Syst., 10 (2011), 169-196. doi: 10.1007/s12346-011-0055-8.  Google Scholar

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.  Google Scholar

[17]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221. doi: 10.1007/s12080-009-0064-2.  Google Scholar

[18]

E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.  Google Scholar

[19]

E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215.  Google Scholar

[20]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback, J. Differential Equations, 255 (2013), 4244-4266. doi: 10.1016/j.jde.2013.08.007.  Google Scholar

[21]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222.  Google Scholar

[22]

J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128. doi: 10.1007/BF01790539.  Google Scholar

[23]

G. Röst, On the global attractivity controversy for a delay model of hematopoiesis, Appl. Math. Comput., 190 (2007), 846-850. doi: 10.1016/j.amc.2007.01.103.  Google Scholar

[24]

G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. doi: 10.1098/rspa.2007.1890.  Google Scholar

[25]

S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications, NATO Sci. Ser. II Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517. doi: 10.1007/1-4020-3647-7_11.  Google Scholar

[26]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260. doi: 10.1007/s002850000070.  Google Scholar

[27]

S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.  Google Scholar

[28]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-94-015-8897-3.  Google Scholar

[29]

A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993. doi: 10.1007/978-94-011-1763-0.  Google Scholar

[30]

D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.  Google Scholar

[31]

A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77. doi: 10.1016/j.mbs.2011.04.004.  Google Scholar

[32]

T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973. doi: 10.1098/rspa.2009.0650.  Google Scholar

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